Multiply the Addition Square

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Age

11 to 14

Challenge level

Thinking mathematically

Mathematical mindsets

Being curious Being collaborative Being resourceful Being resilient

Problem

Take an addition table from $1$ to $10$ (or any other that you like better!):

Now take a $3\times 3$ square of numbers on it, such as this red one:

Now multiply the two diagonally opposite corner numbers together:

And then find the difference between the two answers:

$5 \times 9 = 45$

$7 \times 7 = 49$

$49 - 45 = 4$

Now try the numbers in the green square:

What is the answer?

Investigate other $3\times 3$ squares.

What do you notice?

Can you explain it?

Now try with $2\times 2$ squares and $4\times 4$ squares.

Can you predict your answers? What is happening?

Teachers' Resources

Why do this problem?

This problem is a good one for building on the learners' ability to recognise number properties and reason about numbers. It is easy for all at the start as it only requires simple multiplication and subtraction and gives the satisfaction of finding a pattern. If the results are generalised algebraically it can prove a real challenge.

Possible approach

You could start by posing the problem to the whole group on a computer or using a large-sized addition square.

After this learners could work in pairs on the problem so that they are able to talk through their ideas with a partner. This sheet gives two copies of the addition square which can be used for rough work.

Many learners will tend to rush on and try $2 \times 2$, $4 \times 4$ and other squares and although an interesting pattern can be found doing this, it may stop them from trying to generalise the $3 \times 3$ example. Therefore, it may be wise to stop everyone at an appropriate point and show all who need the help how the numbers can always be named $n, n + 2$ and $n + 4$. (Alternatively, they could be $n - 2$, $n$ and $n + 2$.)

At the end of the lesson learners could discuss the different patterns they have discovered and the various generalisations made.

Key questions

How do the four numbers compare in size?

Try comparing the number in the top left corner of a square with the other three numbers.

So what happens when these numbers are multiplied in the way described?

Can you see a pattern there?

What sort of numbers are these?

Possible extension

Learners could try to generalise, not only the results for $3 \times 3$ squares but also those for all squares on this grid. Alternatively, they could try the same procedure with rectangles rather than squares or use other grids such as multiplication squares.

Possible support

Suggest using a calculator for the multiplication and subtraction or, alternatively, switch to Diagonal Sums which is a similar, but easier, problem.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners